Talagrand's Inequality for a Diagonal Gaussian Measure on Euclidean Space in the Cameron-Martin Cost

In the settings of The Real Numbers: Standing Notation and Background and The Intrinsic Calculus on the Wasserstein Space: Standing Notation, with the dimension $d$ fixed there, the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{d})$ of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces, and Borel maps and functions as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps, let $c=(c_{1},\dots,c_{d})$ be a variance vector, let $\gamma_{c}$ be the diagonal Gaussian measure with variances $c$, and let $\mu$ be a probability measure on $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$ with finite relative entropy $H(\mu\,|\,\gamma_{c})$ with respect to $\gamma_{c}$. For a Borel map $T:\mathbb{R}^{d}\to\mathbb{R}^{d}$ and $k\in\{1,\dots,d\}$ write $T_{k}$ for its $k$-th coordinate function, and $T_{\#}\gamma_{c}$ for the image measure of claim 1 of that lemma.

(Talagrand)¶ There is a Borel map $T:\mathbb{R}^{d}\to\mathbb{R}^{d}$ with $T_{\#}\gamma_{c}=\mu$ such that the nonnegative Borel function $x\mapsto\sum_{k=1}^{d}c_{k}^{-1}\bigl(T_{k}(x)-x_{k}\bigr)^{2}$ is integrable with respect to $\gamma_{c}$ and

$$\int_{\mathbb{R}^{d}}\sum_{k=1}^{d}\frac{\bigl(T_{k}(x)-x_{k}\bigr)^{2}}{c_{k}}\,\gamma_{c}(dx)\le2\,H(\mu\,|\,\gamma_{c}).$$